Monday, January 16, 2017

A Brief Argument on the Prior Probability of Theism

The prior probability of theism: A brief argument for an asymmetry.


In this context, by “theism” I mean “the hypothesis that there is an omnimax agent”, and by “omnimax”, I mean “omnipotent, omniscient, and morally perfect”.

Here, the word “agent” is used in a broad sense, so as not to exclude views on which God is a substance but not a being, or is a trinity rather than a single person, etc.


Let us now introduce some definitions:


V-functions: F is a V-function if F is a function that assigns to different things, situations, states of affairs, etc., real values. In this context, “function” is used broadly; in particular, there is no need for the domain to be a set (it might be bigger than any set).


Moral compatibility for V-functions: If F is a V-function, F is said to be compatible with moral value if the following conditions are met:

MM.1.: If E is morally good, then F(E)>0.

MM.2.: If E is morally bad, then F(E) < 0.

MM.3.: If E1 is morally better than E2, then F(E2) < F(E1).


Agent V-function compatibility: If F is a V-function, and A is an agent, then A is compatible with F if the following conditions are met:

AF1: If F(E)>0, then A values E positively.

AF2: If F(E)<0, then A values E negatively (i. e., A disvalues E).

AF3: If 0 < F(E2) < F(E1), then A positively values E1 more than A positively values E2.

AF4: If 0 > F(E2) > F(E1), then A disvalues E1 more than A disvalues E2.


The introduction of V-functions is not required, but it’s meant to highlight how much morality constrains the potential value structures of a hypothetical agent. In fact, intuitively, there appear to be many more ways in which a V-function can be incompatible with moral value than compatible, in the following sense:

If we are told that F is a V-function, but we’re not told anything else about F and we’re not given any other piece of information, it seems intuitively proper to reckon it’s extremely probable that F is not morally compatible; i. e., the probability that F is morally compatible is close to zero or zero. That’s because there is no particular reason to think that F would agree with morality, with no further info.

If that holds, then it seems to me the following holds for agents, in the following sense:

If A is an agent, F is a V-function, and A is compatible with F (we have no further information about A or F), then it’s extremely probable that F is not morally compatible. But if F is not morally compatible, then (almost certainly, in terms of probability), A is not morally perfect (by the way, for similar reasons, it’s also almost certain that it’s not the case that A is maximally evil).

What if we now introduce the information that A is omnipotent?

Given that power does not seem to make any particular preferences more probable than others in terms of priors, it’s also the case that it’s almost certain that A is not morally perfect.

Clearly, the prior of theism is no greater than the prior of the hypothesis that there is an omnipotent agent. I take that hypothesis to be vastly improbable on its own, in terms of prior probability. But let’s leave that aside. Even assuming that that is not so, we may reason as follows:

Let’s assume that there is some agent A that is omnipotent and is compatible with some V-function F. Then – as sketched above -, it’s almost certain that A is not morally perfect. But if A is not morally perfect, then it is not the case that there is an agent B who is morally perfect and omnipotent. It follows that almost certainly, theism is false.

Alternatively, we can put it in the following manner (here “≈” means that it’s at least very close, in terms of probability):


P(Theism is true)=P(Theism is true, and there is an omnipotent agent)=P(Theism is true, and there is an omnipotent agent compatible with a V-function)+P(Theism is true, and there is an omnipotent agent not compatible with any V-function)P(Theism is true, and there is an omnipotent agent compatible with a V-function)=P(Theism is true There is an omnipotent agent compatible with a V-function) * P(There is an omnipotent agent compatible with a V-function) P(Theism is true There is an omnipotent agent compatible with a V-function) 0.


We just need to justify one more step: The probability P(Theism is true, and there is an omnipotent agent not compatible with any V-function) is almost zero, because if an omnimax agent A exists, there seems to be (clearly) at least one function that assigns real values in a way compatible with As preference structure...but if that’s not so, we can still make a similar argument without using the V-functions, and instead talk about agents whose value structure is fully compatible with moral value.


So, the hypotheses that theism is false has a prior close to 1, and the hypothesis that theism is true, a prior close to 0.

That’s just the prior. Arguments for or against theism might (and would) affect the probabilistic assessments we make. After considering the relevant information available to me to a considerable extent (as always, limited by the availability of time and cognitive limitations), I reckon the probability of theism is still almost zero – in fact, it’s even lower than the prior, given suffering and evil -, but that’s beyond the scope of this post.